I found the answer to this question in Benveniste (Le vocabulaire des institutions indo-européennes, Paris, 1969, I, p.316). To understand the link between phratria and pater, we need to have a glance at the notion of gênos and to the ancient Athenian society in wich 30 gêne were needed to form 1 phratria and 3 phratriai to form 1 phulê. Hence, it's clear that the notion of phratria is linked to one of gênos, the Family (cf. latin gens). So coming from the gênos to end up with the phratry, we move up from the group based on a common birth to one formed of the union of the brothers. Of course they are not proper brothers but they think they have got a common ancestor even a mythical one (cf. for example the gens Iulia whose members (esp. Cesar and Augustus) thought they came from Mars in his union with Rhea Silvia (mother to Romulus and Remus) and from Venus who, united to Anchises, gave birth to Aeneas, himself father to Ascagne, aka Iule founder of Alba.

- How likely is that the festival name 'apatouria' comes from either pater or phratria or both?

According to E. Benveniste (op. cit., I, p.220) "the etymology of apatouria is obvious (sic!). Ancients already translated this word by homopatria, i.e. it's the celebration of those born to the same father".
The initial a- is to be interpreted as a copulative one and not as a privative one ; then, basically, the difference between patouria and patria is just a change of vocalisation (like pater / patros).
Therefore, apatores and phratres are the same for phratres are those born to the same pater. (from P. Chantraine, Dictionnaire Etymologique de la Langue Grecque, Paris, 1968, I, p.96).